Math Forum Discussions

Peter Percival

Posts: 2,623
Registered: 10/25/10

Re: Fundamental period for complex functions
Posted: Aug 12, 2013 11:00 AM

Plain Text

steinerartur@gmail.com wrote:
>
> Suppose f from the complexes to the complexes is periodic. Then, is
> there something like a fundamental period for f?
>
> On the reals, if f is periodic then f may have a fundamental period,
> defined, when it exists, as the smallest period of f. By definition,
> periods are positive, and, if P is the set of periods of f, then the
> fundamental period is p*= infimum P, if p* > 0. In this case, we can
> show p* is in P. So, in this case, p* is the minimum of P. And it's
> known that if f is continuous, periodic and non constant, then f has
> a fundamental period (aka minimum period).
>
> In the complexes, the above definition doesn't make sense, but maybe
> we can define p* as the period with minimum absolute value, provided
> it's positive. So, the definition might be p = minimum {|p| : p is
> period of f}. Does this exist?
>
> For example, 2?i is a period of f(z) = e^z. Does f have a period with
> positive absolute value < 2??
>
> If we define p* for the complexes in this way, then, supposing f is
> continuous, periodic and non constant, then does such p* exist?
>
> Thank you

You may like to read
https://en.wikipedia.org/wiki/Doubly_periodic_function and
https://en.wikipedia.org/wiki/Fundamental_pair_of_periods.

--
Nam Nguyen in sci.logic in the thread 'Q on incompleteness proof'
on 16/07/2013 at 02:16: "there can be such a group where informally
it's impossible to know the truth value of the abelian expression
Axy[x + y = y + x]".